For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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3answers
212 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
2
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0answers
34 views

Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order ...
17
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2answers
524 views

Is the set of quaternions $\mathbb{H}$ algebraically closed?

A skew field $K$ is said to be algebraically closed if it contains a root for every non-constant polynomial in $K[x]$. I know that this is true for $\mathbb{C}$, which is the algebraic closure of ...
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0answers
83 views

Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
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1answer
52 views

limit with quaternion

Let $v\in \mathbb{H}$ and $q:t\in\mathbb{R}\rightarrow q(t)\in \mathbb{H}$. That $q(t) \neq 0$ for all $t\in \mathbb{R}$ and $q^{-1}(t) = \frac{1}{q(t)}$. So don't confuse $q^{-1}(t)$ with inverse ...
2
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1answer
127 views

Algorithm for finding orientation of each face on a polyhedron?

I am working on making a dice rolling application and I need to find out how far in each of the three dimensions I must rotate each of the dice to make the correct side face the camera so the user can ...
0
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1answer
84 views

Quaternion Matrix Multiply

My question is about applying rules of quaternions to quaternion matrices. I know that for some rotation quaternion q = [w, x, y, z], I can find the rotation of ...
1
vote
1answer
92 views

faithful irreducible representation of $A_{4} \times Q_{8}$

Construct a faithful irreducible representation of the group $A_4 \times Q_8$ $A_{4}$ is the alternating group $Q_{8}$ is the quaternions
7
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1answer
186 views

Quaternion group associativity

Consider the eight objects $\pm 1, \pm i, \pm j, \pm k$ with multiplication rules: $ij=k,jk=i,ki=j,ji=-k,kj=-i,ik=-j,i^2=j^2=k^2=-1$, where the minus signs behave as expected and $1$ and $-1$ ...
13
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1answer
209 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...
1
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1answer
200 views

Quaternion exponential map, rotations and interpolation

A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
2
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2answers
383 views

Rotate the axis of rotation of a quaternion by another quaternion

I have two quaternions, $q_1$ and $q_2$. I want to rotate the axis of rotation of $q_1$ by $q_2$. Is there any way of doing this directly, without extracting the axis of rotation from $q_1$, rotating ...
4
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2answers
107 views

Does the square root of $i$ necessitate quaternions?

The square root of i is $\frac{\sqrt{2} + i \sqrt{2}}{2}$. But how is it valid to use a number in expressing the square root of that number?
3
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3answers
213 views

Are there different conventions for representing rotations as quaternions?

I am trying to understand how quaternions are represented as rotations, in particular how to convert from a quaternion representation to a rotation matrix. The following paper by Diebel gives an ...
2
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3answers
913 views

Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
1
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1answer
187 views

Orthogonal procrustes problem using quaternions

Hello I'm trying solve orthogonal procrustes problem in 3d with a help of quaternions. Original problem is: For matrix $A$ find orthogonal matrix $Q$ that $$||A-Q||_F =\min_{\Omega \in SO(3)} ...
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0answers
37 views

“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$

Let a unitary number be one that corresponds to a matrix of the form: $$\left( \begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x) \end{array} \right)$$ This is analogous to ...
6
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3answers
248 views

4 dimensional numbers

I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
1
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1answer
614 views

Coordinate Transformation on Local coordinate system

I am having a point $P(x,y,z)$ in $3D$ with respect to global coordinate system. I want to create an another Local Coordinate System by picking three points $N1, N2, N3$ in 3D. Now I want to know the ...
8
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2answers
295 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
3
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0answers
111 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
2
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0answers
117 views

Understanding quaternions and axis angle representations

I have a sensor that gives me a quaternion. I convert the quaternion to an axis-angle representation using http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToAngle/. When I ...
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2answers
634 views

What does multiplication of two quaternions give?

I'm using quaternions as a means to rotate an object in the application I'm developing. If one quaternion represents a rotation and the second quaternion represents another rotation, what does their ...
0
votes
1answer
75 views

Commutative applying rotations around three axis

Rotating an object in a 3 dimensional space by euler angles might be intuitive but comes with some problems. First, the order of applied rotations around the different axis matters. Second, there is ...
2
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2answers
77 views

Equivalent conditions of quaternion matrix algebra

I am following Theorem 2.3.1 of Maclachlan's and Reid's The Arithmetic of Hyperbolic 3-Manifolds. We define a quaternion algebra $A=\left(\frac{a,b}{F}\right)$ over a field $F$ of characteristic ...
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3answers
2k views

Nullify (zero out, cancel) rotation in an arbitrary axis in a Quaternion

Question: How do you nullify (zero out) rotation around an arbitrary axis in a Quaternion? Example: Let's say you have an object with quaternion orientation $A$. You also have a rotation quaternion ...
3
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1answer
72 views

Some questions about quaternions.

It is possible make something like complexification of a real vector space using quaternions? If yes, it's similar to complex case or there are considerable differences? Has been studied a quaternion ...
2
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2answers
309 views

Proof that Quaternion Algebras are simple

I have a proof that every quaternion algebra over a field $A=\left(\frac{a,b}{F}\right)$ is simple, i.e. has no nontrivial two-sided ideals, which appeals to the algebraic closure of $F$ and the ...
2
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3answers
177 views

Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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0answers
49 views

Optimimal rotation using non-linear conjugate gradient

The problem I'd like to ask is the following : let $M_1$ and $M_2$ two rigid bodies with a quadratic constraint function $f$ attached to its grid points. $M_2$ is always kept static while $M_1$ can be ...
4
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2answers
2k views

Combining rotation quaternions

If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis. The the order of rotation ...
5
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2answers
510 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
0
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1answer
35 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
4
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3answers
318 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
3
votes
1answer
195 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
3
votes
2answers
59 views

What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ...
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1answer
45 views

How does one derive this rotation quaternion formula?

given an angle and an axis, the corresponding quaternion can be computed like this. $w = \cos( Angle/2)$ $x = \text{axis}.x * \sin( Angle/2 )$ $y = \text{axis}.y * \sin( Angle/2 )$ $z = ...
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0answers
166 views

How is the Quaternion multiplication derived?

Quaternion multiplication seems suspiciously similar to the cross product. How is it derived? Here is a description of the multiplication: Let $Q_1$ and $Q_2$ be two quaternions, which are defined, ...
3
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3answers
131 views

Quaternions: Difference(s) between $\mathbb{H}$ and $Q_8$

What is the difference between $\mathbb{H}$ and $Q_8$? Both are called quaternions.
0
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1answer
186 views

How do I calculate a position in front of a quaternion given the initial position and the quaternion?

Well I want to determine the position in front of, lets say 'an object'. 'An objects' has a position (pX,pY,pZ) and a quaternion rotation (qX,qY,qZ,qW) (where qW seems always to be ...
0
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1answer
166 views

How are these formulas for Quaternion -> Rotation Matrix related?

I'm trying to write a program to convert a quaternion to a rotation matrix. One source I found is: ...
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1answer
119 views

Quaternions, torque, and impulse.

In a physics simulation I have a solid ball of mass $m$ and moment of interia $M$ (which is a diagonal matrix with all entries equal to ${2\over5}mr^2=i$). Its instantaneous rotation is given by a ...
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3answers
574 views

half sine and half cosine quaternions

Something is a little bit unclear to me. In the image below you see that you need to divide the angle by a half. Acccording to wikipedia they say that this is so that I could rotate clockwise or ...
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4answers
1k views

Quaternions: why does ijk = -1 and ij=k and -ji=k

Currently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: ...
2
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2answers
304 views

Detail in the proof of the quaternion rotation identity

I am trying to understand the proof of the quaternion rotation identity illustrated in wikipedia ...
6
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2answers
207 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
2
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1answer
261 views

Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
2
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1answer
79 views

How can I break down a rotation of known amount around a known axis into two rotations of unknown amounts around known axes?

I have a vector that's been rotated a known amount about a known axis. I would like to break this rotation down into two separate rotations around known, linearly independent axes where the amounts I ...
4
votes
1answer
139 views

How to Perform Quaternion Multiplication

Everywhere that I've looked, it seems to be assumed that $i^{2} = j^{2} = k^{2} = - 1$, along with the other rules of quaternion multiplication. However - for my homework - I'm being asked to show ...
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1answer
103 views

inner product on matrices with quaternionic entries

let $\mathbb{H}$ be the quaternions and $Mat(n,\mathbb{H})$ be the vector space of $n\times n$ matrices over $\mathbb{H}$. Let $H(n,\mathbb{H}):=\{ A\in Mat(n,\mathbb{H}): \overline{A}^t=A \}$ be the ...